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Questions tagged [monomorphisms]

This tag is for questions related to monomorphisms.

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1answer
32 views

Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
2
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1answer
52 views

Monomorphisms preserved in a pullback

The question is answered, e.g, here. Suppose $(P, p_1, p_2)$ is a pullback for $f:A\to C$ and $g:B\to C$ with $fp_1 = gp_2$. if $f$ is a monomorphism, show $p_2$ is a monomorphism. What we do is ...
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4answers
43 views

Counterexample: functor does not preserve monomorphisms

I need to show that functors need not preserves mono's and epi's. For epi's, I have as counterexample the forgetful functor $F : \mathbf{Ring} \to \mathbf{Set}$. We have that $f: \mathbb{Z} \...
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0answers
61 views

In R-Mod, all monomorphisms are equalizers

I want to prove that in the category of R-Modules, all monomorphisms are equalizers. We start by assuming that if $f : A \to B$ is mono and if $f$ equalizes $g : B \to C$ and $h : B \to C$ (i.e. $ g \...
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1answer
49 views

Are all split epimorphisms effective? [duplicate]

An epimorphism is called split if it has a section (a right inverse). An epimorphism is called effective if it has a kernel pair and is the coequalizer of its kernel pair. The ncatlab implies here ...
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1answer
54 views

Every $\lambda-$pure morphism in a locally $\lambda-$presentable category is a regular monomorphism

Consider the page from the book by Adamek & Rosicky: Locally presentable and accessible categories. given below. I need to derive this: Here is the statement (2) to prove the universal ...
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0answers
40 views

Proof verification: An arrow which is monic under a faithful functor is itself monic

Context: To introduce some symbols and such, what I'm seeking to prove is this: Let $F$ be a faithful functor. Suppose $F(f)$ is a monic arrow. Show $f$ is monic. This came up as part of a class ...
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1answer
26 views

Investigate whether the given transformation is a monomorphism / epimorphism. Find image and kernel

I have serious doubts - I will be very grateful if someone will help me here Investigate whether the given transformation is a monomorphism / epimorphism. Find its image and kernel. $$ F \in L(\...
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1answer
89 views

$\lambda$-pure morphisms in $\lambda$-accessible categories are monos, unclear proof

This is Proposition 2.29 from the book Locally Presentable and Accessible Categories by Jiří Adámek and Jiří Rosický. Above is a proof that $\lambda$-pure morphisms in $\lambda$-accessible categories ...
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1answer
48 views

Is the unique subobject also mono?

Suppose $u : S → A$ and $v : T → A$ with codomain $A$ are monomorphisms, we write $u ≤ v$ if $u$ factors through $v$—that is, if there exists $φ : S → T$ such that $u = v ∘ φ$. I understand that ...
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1answer
19 views

Relation between homomorphisms and monomorphims of finite groups

For any finite groups L and G, let $h(L,G)$ denote the number of homomorphisms from L to G and $i(L,G)$ denote the number of monomorphisms from L to G. Proof that $$h(L,G)=\sum_{N \triangleleft \ L} ...
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1answer
67 views

Existence of morphisms in a free completion under directed colimits,$\lambda$-accessible category

Let $\cal K$ be a $\lambda$-accessible category with directed colimits and $\cal C$ be its representative full subcategory consiting of $\lambda$-presentable objects. Let $\cal L$ be free completion ...
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1answer
32 views

Monomorphisms, unclear basic property, Functor

Suppose that for morphisms in a category it holds that $f\circ u=v\circ f'$ and $g\circ u=v\circ g'$ and that for a functor $F$, $Fv$ is a monomorphism. Suppose that $Ff'$ and $Fg'$ are distinct. WHY ...
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0answers
58 views

A question about linearly independent monomorphisms.

My question was too big to put it in the title: Having a set of linearly independent field monomorphisms $K→L$, is it still possible to express one of these monomorphisms as a linear combination of ...
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2answers
209 views

How exactly is a Field monomorphism defined?

Multiple times in my question I'll use the term 'a mapping of $x$', what I mean with this is one of the element to which $x$ can be mapped to (since there could be more than one). We have Two fields, ...
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3answers
112 views

Is an injective function always monomorphic

Is an injective function monomorphic (in category of sets)? How do I prove this? Conversely, I could prove that a monomorphic function is injective in the following manner: If f is monic, then $f\...
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2answers
67 views

Is there a counter example for monmorphism

Monomorhism as defined is: A morphism $f: A \to B$ is a monomorphism if for every object $C$ and every pair of morphisms $g,h: C \to A$ the condition $f\circ g = f\circ h$ implies $g = h$. Applying ...
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1answer
44 views

Plain monomorphism,the defintion

In several places the notions of a plain monomorphism or a plain epimorphism are used, but never defined. See eg. here on page 60, or search google with query plain monomorphisms tholen Do they ...
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2answers
74 views

Confused about the definition of a Field Extension

A particular author defines a Field Extension as a monomorphism (to be more detailed, as an injective homomorphism) between two Fields. However, my idea of a Field Extension is that of a pair of ...
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1answer
77 views

A counterexample to Cantor-Schroeder-Bernstein in groups.

We need to find a counterexample to the Cantor-Schroeder-Bernstein theorem for the category of groups (that is, find two monomorphisms $\varphi\colon G\to H$ and $\psi\colon H\to G$ such that $G$ and $...
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0answers
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Mono(epi)morphisms as im(sub)mersions in the category of manifolds

In the definition of the category of smooth manifolds it's usual to take the morphisms to be simply smooth functions, and this results in having monomorfisms and epimorphisms to be injective and ...
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3answers
104 views

How do kernels prove how a function fails to be injective

I know that for $f\colon X \to Y$, where $e_Y$ is the identity of $Y$: $$ \ker(f) = \left\{x \in X \, \middle| \, f(x) = e_Y \right\} $$ I've learnt that kernels imply how much a homomorphism fails ...
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2answers
82 views

Is $\mathbb Z_7$ an extension of $\mathbb Z_2$?

Is $\mathbb Z_7$ a extension of $\mathbb Z_2?$ I know it is not, because we can't establish a monomorphism between $\mathbb Z_7$ and $\mathbb Z_2$,i.e. it doesn't exist an injective homomorphism ...
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2answers
524 views

Any category with zero object, every kernel is monomorphism.

This is the problem 5.53 of Rotman, Homological Algebra. In any category having a zero object calling it $0$, prove that every kernel is monomorphism. Let $f:A\to B$ and $\ker(f):K\to A$ be the ...
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1answer
101 views

Prove linear transformation is monomorphism

I have troubles with following exercise: $(X, \langle \cdot, \cdot \rangle_X)$ and $(Y, \langle \cdot, \cdot \rangle_Y)$ are euclidean spaces. Linear transformation $f \in L(X, Y)$ has property $$\...
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1answer
41 views

How to prove that a partition of a set is a monomorphism in the category of sets

I am having some trouble proving exercise 4.23 in the book "Sets for Mathematics" by Lawvere and Rosebrugh. The problem is to prove that from a partition $p:A \rightarrow I$, which is defined there ...
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1answer
93 views

How are the monomorphisms defined?

Let $C$ be an algebraic closure of $F$, let $f\in F[x]$ be irreduccible and $a,b\in C$ the roots of $f$. We have the following theorem: If $E$ is an algebraic extension of $F$, $C$ is an ...
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1answer
55 views

How could we show the equality?

We have that $C$ is an algebraic closure of $F$ and $a,b\in C$. We have the $F$-monomorphism $\tau: C\hookrightarrow C$ with $\tau (a)=b$. From the mapping we have that $\tau (C)\subseteq C$. ...
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1answer
67 views

Does every arrow in a thin category neccessarily have to be monic and epic by default?

In other words, will there ever be an arrow in a thin category that is not either monic or epic?
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1answer
336 views

Pushouts in the category of sets preserve monomorphisms

Can anyone give me a "hands on" proof of why the pushout of an injective function (along any other function) is again an injective function? I know sets form a topos and toposes are adhesive, but is ...
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0answers
89 views

Separable extension and monomorphism

Let $F$ and $K$ be two fields, $F$ is a finite extension of $K$, $[F:K]=n$. If there exist exactly $n$ monomorphisms $\sigma_1,\sigma_2,..,\sigma_n$ $$\sigma_i: F\rightarrow \overline{K}$$ where $\...
2
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1answer
64 views

Algebraic closures and monomorphisms

Let $F$ be a field and $f\in F[X]$, an irreducible polynomial over $F$, $C$ an algebraic closure of $F$ and $a,b \in C$ two roots of $f$. In an exercise I proved the existence of an $F-$monomorphism $...
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1answer
212 views

Proof that f is a monomorphism [closed]

let $f:A \rightarrow B$ and $g:B \rightarrow A$ given that $\;\;g \circ f = 1_A$ how can I prove that f is a monomorphism? (I'm used to prove monomorphism in a different way, so I'm kind of lost)
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0answers
55 views

Proof that monomorphism is injective [duplicate]

In category theory, a monomorphism is a situation where $g: A \rightarrow B \;\;$ , $\;\;h: A \rightarrow B\;\;$ and $\;\;f:B \rightarrow C$ $f \circ g = f \circ h \;\text{ imples in } \; g = ...
2
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1answer
785 views

Monomorphisms, epimorphisms and isomorphisms of groups category

I would like to solve the Problem 2.12 in Szekeres, A Course in Modern Mathematical Physics: Show that the class of groups as objects with homomorphisms between groups as morphisms forms a ...
2
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1answer
65 views

Does this property really characterize monomorphisms?

In the post Does this property characterize monomorphisms?, I do not see how the third condition is equivalent to the others. Specifically, I require that $k_0$ and $k_1$ be isomorphisms in order that ...
0
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1answer
346 views

$K$-monomorphism that is not $K$-automorphism?

I am confused by the terminology where $K$ precedes terms such as $K$-monomorphism and $K$-automorphism in Galois theory. I am trying to come up with a simple example about $K$-monomorphism that is ...
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2answers
458 views

How many monomorphisms are there $\mathbb{Q} \rightarrow \mathbb{C}$

I was surprised as to how no sources online really took this particular monomorphism for an example while it seemed very common. My thoughts are that it is only the identity map that exists, so $1$. ...
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1answer
186 views

Proofs in Category Theory: Monicity of Composition of Monic Arrows

I've just started working on proofs in category theory and I'm interested in thinking about the structure of a proof in terms of this domain. This means thinking about the proof in terms of arrows ...
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1answer
24 views

Linear Mapping Monorphism

The question I have reads as follows: Prove the following statement: A linear map $f: V \to W$ is a monomorphism if and only if f(S) is Linearly Independent in W whenever S is Linearly Indepdendent ...
2
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1answer
159 views

Field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$.

List (with proof) all field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$. So I can see that the field monomorphism from $\mathbb{Q}(\sqrt{5})$ to $\mathbb{C}$ are $p+q\sqrt{5}$ $p-q\...
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1answer
52 views

Categorical way of making monos commute

Is there a categorical way (in terms of diagrams, limits, lifting properties etc) to formulate the requirement that for every pair of monos $f,g:C \to D$ there should be an endomorphism $h:D \to D$, s....
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1answer
415 views

Are Monomorphisms injective?

In the categories of topological spaces, rings, groups and sets I know that a morphism is a monomorphism iff it's injective. Things are different for schemes. In fact I know that a scheme injective ...
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0answers
165 views

Monomorphisms and injectivity predicates

This is a curiosity question which struck me at a less-than-optimal moment; I apologize for not having thought much about it. Motivation. It is well-known that monomorphisms in a concrete category ...
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0answers
354 views

Monomorphism preservation by pullback

I'm working through Category theory for the sciences (by David Spivak) and I'm a bit stuck in section 2 - my problem is with Proposition 2.7.5.5 and Exercise 2.7.5.6, I hope someone can help :) The ...
4
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1answer
147 views

Does this property characterize monomorphisms?

Is requiring that $f:\mathrm{Hom}(A,B)$ is mono the same as requiring that the pullback $A\times_BA$ of $f$ along itself is isomorphic to $A$? In Sheaves in Geometry and Logic, I read the following ...
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1answer
62 views

Not understanding a line in a proof concerning Monomorphism and injectivity

In the proof that "in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective" given in Wikipedia http://en.wikipedia.org/...